In 1150, it was recognized in England that coins should have a standard weight of precious metal as the basis for their value. A guinea, for example, was supposed to contain 128 grains of gold. (There are 360 grains in an ounce.) In the “trial of the pyx,” coins minted under contract to the Crown were weighed and compared to standard coins (which were kept in a wooden box called the pyx). Coins were allowed to deviate by no more than 0.28 grains—roughly equivalent to specifying that the standard deviation should be no greater than 0.09 grains (although they didn’t know what a standard deviation was in 1150). In fact, the trial was performed by weighing 100 coins at a time and requiring the sum to deviate by no more than 100 × 0.28 = 28 or 28 grains—equivalent to the sum having a standard deviation of about 9 grains.

a) In effect, the trial of the pyx required that the mean weight of the sample of 100 coins have a standard deviation of 0.09 grains. Explain what was wrong with performing the trial in this manner.

b) What should the limit have been on the standard deviation of the mean?